Colombeau had constructed his well-known algebras by algebraic
methods. No topology had appeared in his construction.
Our aim is to give a purely topological description of Colombeau
type algebras. We show that such algebras fit very well in the general
theory of the well known sequence spaces forming appropriate algebras.
All these classes of algebras are simply determined by
the (locally convex) space and a sequence of weights
which serves to construct an ultrametric on the
sequence space .
The sequence
is assumed to be decreasing to zero. This implies
that sequence spaces under consideration (
) contain as a
subspace
and that they induce the discrete
topology on .
Our analysis implies that if one has a Colombeau
type algebra containing the Dirac delta distribution as an
embedded Colombeau generalized function, then the topology induced on
the basic space must be discrete. This is an analogous result to the
Schwartz's ``imposibility result'' concerning the product of
distributions.
A major part of the talk
is devoted to embeddings of ultradistribution and hyperfunction spaces
into corresponding classes of sequence spaces.